Theorem imbi12 336

Description: Closed form of imbi12i 340. Was automatically derived from its "Virtual Deduction" version and Metamath's "minimize" command. (Contributed by Alan Sare, 18-Mar-2012.)

Assertion

Ref	Expression
imbi12	|- ( ( ph <-> ps ) -> ( ( ch <-> th ) -> ( ( ph -> ch ) <-> ( ps -> th ) ) ) )

Proof of Theorem imbi12

Step	Hyp	Ref	Expression
1		simplim 163	. . 3 |- ( -. ( ( ph <-> ps ) -> -. ( ch <-> th ) ) -> ( ph <-> ps ) )
2		simprim 162	. . 3 |- ( -. ( ( ph <-> ps ) -> -. ( ch <-> th ) ) -> ( ch <-> th ) )
3	1, 2	imbi12d 334	. 2 |- ( -. ( ( ph <-> ps ) -> -. ( ch <-> th ) ) -> ( ( ph -> ch ) <-> ( ps -> th ) ) )
4	3	expi 161	1 |- ( ( ph <-> ps ) -> ( ( ch <-> th ) -> ( ( ph -> ch ) <-> ( ps -> th ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 197

This theorem is referenced by:  imbi12i  340  bj-imbi12  32567  ifpbi12  37833  ifpbi13  37834  imbi13  38726  imbi13VD  39110  sbcssgVD  39119